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摘要:
为了探讨花朵授粉算法（FPA）在解算多模函数优化问题中存在的不足，通过定义种群多样性及差异性指标，定性分析了FPA在多模复杂函数优化中的寻优缺点。基于模拟退火思想优化全局授粉过程，并利用Nelder-Mead单纯形搜索技术对花朵局部授粉进行重构，提出一种新的花朵授粉寻优架构。仿真结果表明，相对于基本的FPA、布谷鸟算法、萤火虫算法，改进花朵授粉算法能够有效避免陷入局部最优，具备优异的全局勘探和局部开采能力，对多模优化问题具有一定优势。
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关键词:
- 花朵授粉算法(FPA) /
- 模拟退火 /
- Nelder-Mead单纯形法 /
- 多模函数优化
Abstract:In order to discuss the defects of flower pollination algorithm (FPA) in solving multimodal optimization problems, the optimal disadvantages of flower pollination algorithm in multimodal function optimization were qualitatively analyzed by defining population diversity and difference index. And then a new framework of FPA was constructed by optimizing the global pollination process based on the simulated annealing idea and using Nelder-Mead simplex search method to reconstruct the local pollination process. The simulation results show that the improved flower pollination algorithm can effectively avoid falling into local optimum and has better global exploration and local exploitation abilities, which has advantages to solve multimodal function optimization, compared with primary flower pollination algorithm, cuckoo search algorithm and firefly algorithm.
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图 2 Sphere函数FPA寻优参数变化曲线
Figure 2. Parameter variation curves of Sphere function using FPA optimization
图 3 Sphere函数FPA寻优花粉分布
Figure 3. Pollen distribution of Sphere function using FPA optimization
图 6 Bridge函数FPA寻优参数变化曲线
Figure 6. Parameter variation curves of Bridge function using FPA optimization
图 7 Bridge函数FPA寻优花粉分布
Figure 7. Pollen distribution of Bridge function using FPA optimization
图 10 Sphere函数NS-FPA寻优参数变化曲线
Figure 10. Parameter variation curves of Sphere function using NS-FPA optimization
图 11 Sphere函数NS-FPA寻优花粉分布
Figure 11. Pollen distribution of Sphere function using NS-FPA optimization
图 13 Bridge函数NS-FPA寻优参数变化曲线
Figure 13. Parameter variation curves of Bridge function using NS-FPA optimization
图 14 Bridge函数NS-FPA寻优花粉分布
Figure 14. Pollen distribution of Bridge function using NS-FPA optimization
图 16 F11函数50次寻优结果对比
Figure 16. Comparison of 50 times' optimization results of F11 function
表 1 4个算法的参数设置
Table 1. Parameter setup for four algorithms
算法 参数设置 FPA 转换概率p=0.8；参数λ=1.5 CS 发现概率pa=0.25；步长调节量α=0.01；
参数β=1.5FA 步长因子α=0.25；吸引度因子β=0.2；
光强吸收强度γ=1NS-FPA 转换概率p=0.8；温度衰减系数γ=0.7；
初温概率p0=0.8；参数λ=1.5注：公共参数:种群规模N=50；最大迭代次数Kmax，F1~F13取100，F14~F16取10 000。 
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表 2 16个标准测试函数
Table 2. 16 standard test functions
代号 函数 表达式 解空间 最小值 F1 Sphere 
[-100, 100]2 0 F2 Brid 
[-2π, 2π]2 -106.764 5 F3 Roots 
[-2, 2]2 -1 F4 Bohachevsky 
[-100, 100]2 0 F5 Eggcrate 
[-10, 10]2 0 F6 Guichi-f4 
[-2, 2]2 -3.253 9 F7 Cross-in-Tray 
[-10, 10]2 -2.062 61 F8 Holder Table 
[-10, 10]2 -19.208 5 F9 Drop-Wave 
[-5.12, 5.12]2 -1 F10 Levy 
[-10, 10]2 0 F11 Bridge 
[-10, 10]2 -3.005 4 F12 Shubert 
[-50, 50]2 -186.730 9 F13 Rastrigin 
[-20, 20]4 0 F14 Ackley 
[-32, 32]30 0 F15 Griewank 
[-600, 600]40 0 F16 Vincent 
[0.25, 10]50 -50
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表 3 16个标准测试函数优化结果对比
Table 3. Comparison of optimization results of 16 standard test functions
函数代号 FPA CS 最优值 平均值 标准差 最优值 平均值 标准差 F1 2.82 5.42 1.63 9.85×10-7 2.37×10-3 5.49×10-4 F2 -106.764 5 -106.762 7 2.24×10-3 -106.764 5 -106.757 0 1.64×10-3 F3 -0.999 8 -0.997 0 1.88×10-3 -0.999 7 -0.994 5 1.21×10-3 F4 6.46×10-5 2.46×10-2 3.49×10-2 2.56×10-1 1.81 3.97×10-1 F5 2.09×10-6 2.41×10-3 5.07×10-3 1.88×10-3 0.53 1.24×10-1 F6 -3.253 9 -3.253 7 3.20×10-4 -3.253 9 -3.253 3 1.29×10-4 F7 -2.062 6 -2.062 3 7.46×10-5 -2.062 6 -2.062 5 2.11×10-5 F8 -19.208 5 -19.208 3 1.73×10-4 -19.208 5 -19.208 2 7.18×10-5 F9 -0.993 3 -0.947 3 1.77×10-2 -0.986 5 -0.936 2 1.77×10-2 F10 1.24×10-5 2.93×10-2 3.70×10-2 4.06×10-6 5.60×10-3 1.40×10-3 F11 -3.004 1 -2.877 1 8.56×10-2 -2.928 0 -2.651 2 7.64×10-2 F12 -186.730 7 -186.712 5 2.68×10-2 -186.730 8 -186.709 8 6.13×10-3 F13 6.60 15.36 5.01 2.93 8.05 1.46 F14 3.87 6.34 1.17 4.66 8.35 0.92 F15 4.64 11.29 2.66 2.73 6.97 1.21 F16 -49.61 -49.22 0.16 -49.84 -49.74 9.33×10-2 F1 8.63×10-11 5.79×10-9 5.66×10-9 3.88×10-9 1.82×10-5 2.89×10-5 F2 -106.764 5 -104.939 3 5.23 -106.764 5 -106.764 5 4.76×10-6 F3 -1.000 0 -0.999 8 1.06×10-4 -1.000 0 -0.999 9 7.81×10-5 F4 6.70×10-7 4.88×10-4 8.66×10-4 8.32×10-9 5.00×10-5 3.94×10-5 F5 8.34×10-8 3.59×10-6 5.29×10-6 5.15×10-9 1.88×10-7 1.70×10-7 F6 -3.253 9 -3.253 1 8.20×10-4 -3.253 9 -3.253 9 5.93×10-6 F7 -2.062 6 -2.062 6 1.77×10-4 -2.062 6 -2.062 6 1.30×10-8 F8 -19.208 5 -19.207 9 9.73×10-4 -19.208 5 -19.208 5 4.50×10-10 F9 -1.000 0 -0.994 9 1.75×10-2 -1.000 0 -0.999 9 1.99×10-4 F10 4.68×10-9 1.41×10-7 2.10×10-7 1.26×10-10 6.79×10-9 7.70×10-9 F11 -3.005 4 -2.933 6 0.20 -3.005 4 -3.005 1 3.18×10-4 F12 -186.730 9 -177.764 8 13.30 -186.730 9 -186.730 8 5.50×10-4 F13 2.38×10-6 1.33 1.41 0 1.18×10-7 7.42×10-7 F14 1.80×10-3 3.83×10-3 1.12×10-3 6.46×10-12 1.62×10-7 1.09×10-6 F15 3.00×10-4 7.47×10-2 0.19 2.97×10-4 1.32×10-3 7.53×10-4 F16 -49.87 -9.62 9.83×10-2 -50.00 -49.99 2.51×10-4
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表 4 不同p0、γ参数下的NS-FPA寻优结果
Table 4. Optimization results of NS-FPA with different p0 and γ
p0 平均值 标准差 γ 平均值 标准差 0.1 -0.998 08 9.20×10-3 0.1 -0.996 70 1.28×10-2 0.2 -0.998 44 9.19×10-3 0.2 -0.999 11 4.04×10-3 0.3 -0.998 61 8.57×10-3 0.3 -0.999 15 3.27×10-3 0.4 -0.999 14 4.14×10-3 0.4 -0.999 65 1.40×10-3 0.5 -0.999 35 2.73×10-3 0.5 -0.999 72 1.21×10-3 0.6 -0.999 81 9.14×10-4 0.6 -0.999 78 1.14×10-3 0.7 -0.999 99 6.52×10-5 0.7 -0.999 83 9.39×10-4 0.8 -0.999 93 9.59×10-5 0.8 -0.999 98 7.47×10-5 0.9 -0.999 86 7.12×10-4 0.9 -0.999 89 6.15×10-4
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表 5 NS-FPA与FPA的平均消耗时间对比
Table 5. Comparison of mean consumption time between NS-FPA and FPA
函数代号 平均耗时/s NS-FPA与FPA平均
耗时的百分比NS-FPA FPA F5 0.28 0.20 142.56 F6 0.28 0.20 142.76 F11 0.31 0.22 137.15 F13 0.28 0.22 126.89 F14 33.56 27.22 123.31 F15 34.99 28.63 122.24
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